Question: Vanessa is 3 times as old as Brandon. Ten years ago, Vanessa was 8 times as old as Brandon. How old is Vanessa now?
Answer: We can use the given information to write down two equations that describe the ages of Vanessa and Brandon. Let Vanessa's current age be $v$ and Brandon's current age be $b$ The information in the first sentence can be expressed in the following equation: $v = 3b$ Ten years ago, Vanessa was $v - 10$ years old, and Brandon was $b - 10$ years old. The information in the second sentence can be expressed in the following equation: $v - 10 = 8(b - 10)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $v$ , it might be easiest to solve our first equation for $b$ and substitute it into our second equation. Solving our first equation for $b$ , we get: $b = v / 3$ . Substituting this into our second equation, we get: $v - 10 = 8($ $(v / 3)$ $- 10)$ which combines the information about $v$ from both of our original equations. Simplifying the right side of this equation, we get: $v - 10 = \dfrac{8}{3} v - 80$ Solving for $v$ , we get: $\dfrac{5}{3} v = 70$ $v = \dfrac{3}{5} \cdot 70 = 42$.